Winter model in the quasi-continuum limit

We study Winter or delta-shell model at finite volume (length), describing a small resonating cavity weakly-coupled to a large one. For generic values of the coupling, a resonance of the usual model corresponds, in the finite-volume case, to a compression of the spectral lines; for specific values of the coupling, a resonance corresponds instead to a degenerate or a quasi-degenerate doublet. A secular term of the form g^3 N occurs in the perturbative expansion of the momenta (or of the energies) of the particle at third order in g, where g is the coupling among the cavities and N is the ratio of the length of the large cavity over the length of the small one. These secular terms, which tend to spoil the convergence of the perturbative series in the large volume case, N >> 1, are resummed to all orders in g by means of standard multi-scale methods. The resulting improved perturbative expansions provide a rather complete analytic description of resonance dynamics at finite volume.